Generalized Entropies

TL;DR

This paper introduces a generalized entropy measure for quantum systems based on hypothesis testing, formulated as a semidefinite program, and explores its properties, bounds, and operational implications.

Contribution

It proposes a new entropy measure for quantum systems, extending von Neumann and Shannon entropies, with a semidefinite programming formulation and derived bounds.

Findings

Established basic properties of the generalized entropy

Derived upper and lower bounds using smooth entropies

Proved a chain rule for the entropy based on hypothesis test decomposition

Abstract

We study an entropy measure for quantum systems that generalizes the von Neumann entropy as well as its classical counterpart, the Gibbs or Shannon entropy. The entropy measure is based on hypothesis testing and has an elegant formulation as a semidefinite program, a type of convex optimization. After establishing a few basic properties, we prove upper and lower bounds in terms of the smooth entropies, a family of entropy measures that is used to characterize a wide range of operational quantities. From the formulation as a semidefinite program, we also prove a result on decomposition of hypothesis tests, which leads to a chain rule for the entropy.